Problem CSR 04 Ex1 GM99

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex1 GM99

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex1 GM99

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(a(), b(), X) -> f(X, X, X)
     , c() -> a()
     , c() -> b()}

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: f^#(a(), b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1()
              , 3: c^#() -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(a(), b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}
             
             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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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: {c^#() -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_1() = [0]
                        [3]
                        [3]
           
           * 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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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: {c^#() -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_2() = [0]
                        [3]
                        [3]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a(), b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1()
              , 3: c^#() -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
                                  [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c^#() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(a(), b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}
             
             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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c^#() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2() = [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: {c^#() -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c^#() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2() = [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: {c^#() -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                c_2() = [0]
                        [1]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a(), b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1()
              , 3: c^#() -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c^#() = [0]
                c_1() = [0]
                c_2() = [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(a(), b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}
             
             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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1() = [0]
                c_2() = [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: {c^#() -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                c_1() = [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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1() = [0]
                c_2() = [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: {c^#() -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                c_2() = [0]
    
    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 TimeUnknown
Answer
MAYBE
InputCSR 04 Ex1 GM99

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex1 GM99

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(a(), b(), X) -> f(X, X, X)
     , c() -> a()
     , c() -> b()}

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: f^#(a(), b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1()
              , 3: c^#() -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(a(), b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}
             
             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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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: {c^#() -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_1() = [0]
                        [3]
                        [3]
           
           * 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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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: {c^#() -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_2() = [0]
                        [3]
                        [3]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a(), b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1()
              , 3: c^#() -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
                                  [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c^#() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(a(), b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}
             
             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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c^#() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2() = [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: {c^#() -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c^#() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2() = [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: {c^#() -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                c_2() = [0]
                        [1]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a(), b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1()
              , 3: c^#() -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c^#() = [0]
                c_1() = [0]
                c_2() = [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(a(), b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}
             
             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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1() = [0]
                c_2() = [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: {c^#() -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                c_1() = [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(f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1() = [0]
                c_2() = [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: {c^#() -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                c_2() = [0]
    
    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.