Problem Transformed CSR 04 Ex24 Luc06 GM

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex24 Luc06 GM

stdout:

MAYBE

Problem:
 a__f(b(),X,c()) -> a__f(X,a__c(),X)
 a__c() -> b()
 mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
 mark(c()) -> a__c()
 mark(b()) -> b()
 a__f(X1,X2,X3) -> f(X1,X2,X3)
 a__c() -> c()

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex24 Luc06 GM

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex24 Luc06 GM

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  a__f(b(), X, c()) -> a__f(X, a__c(), X)
     , a__c() -> b()
     , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
     , mark(c()) -> a__c()
     , mark(b()) -> b()
     , a__f(X1, X2, X3) -> f(X1, X2, X3)
     , a__c() -> c()}

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: a__f^#(b(), X, c()) -> c_0(a__f^#(X, a__c(), X))
              , 2: a__c^#() -> c_1()
              , 3: mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
              , 4: mark^#(c()) -> c_3(a__c^#())
              , 5: mark^#(b()) -> c_4()
              , 6: a__f^#(X1, X2, X3) -> c_5()
              , 7: a__c^#() -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{3}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                `->{6}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}->{6}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 {3}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
                  , a__f^#(X1, X2, X3) -> c_5()
                  , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                  , mark(c()) -> a__c()
                  , mark(b()) -> b()
                  , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                  , a__c() -> b()
                  , a__f(X1, X2, X3) -> f(X1, X2, X3)
                  , a__c() -> c()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [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: {mark^#(c()) -> c_3(a__c^#())}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                      [2]
                a__c^#() = [2]
                           [2]
                           [0]
                mark^#(x1) = [0 2 2] x1 + [7]
                             [2 2 0]      [7]
                             [0 0 0]      [7]
                c_3(x1) = [2 2 0] x1 + [1]
                          [0 0 0]      [7]
                          [2 0 0]      [3]
           
           * Path {4}->{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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [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: {a__c^#() -> c_1()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [6]
                      [2]
                a__c^#() = [2]
                           [2]
                           [2]
                c_1() = [1]
                        [0]
                        [0]
                mark^#(x1) = [2 0 2] x1 + [7]
                             [0 2 0]      [3]
                             [2 0 0]      [7]
                c_3(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]
           
           * Path {4}->{7}: 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [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: {a__c^#() -> c_6()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [6]
                      [2]
                a__c^#() = [2]
                           [2]
                           [2]
                mark^#(x1) = [2 0 2] x1 + [7]
                             [0 2 0]      [3]
                             [2 0 0]      [7]
                c_3(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]
                c_6() = [1]
                        [0]
                        [0]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [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: {mark^#(b()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [2]
                      [2]
                      [2]
                mark^#(x1) = [0 2 0] x1 + [7]
                             [2 2 0]      [3]
                             [2 2 2]      [3]
                c_4() = [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: a__f^#(b(), X, c()) -> c_0(a__f^#(X, a__c(), X))
              , 2: a__c^#() -> c_1()
              , 3: mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
              , 4: mark^#(c()) -> c_3(a__c^#())
              , 5: mark^#(b()) -> c_4()
              , 6: a__f^#(X1, X2, X3) -> c_5()
              , 7: a__c^#() -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{3}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                `->{6}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}->{6}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 {3}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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:
                 {  mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
                  , a__f^#(X1, X2, X3) -> c_5()
                  , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                  , mark(c()) -> a__c()
                  , mark(b()) -> b()
                  , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                  , a__c() -> b()
                  , a__f(X1, X2, X3) -> f(X1, X2, X3)
                  , a__c() -> c()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6() = [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: {mark^#(c()) -> c_3(a__c^#())}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                a__c^#() = [0]
                           [2]
                mark^#(x1) = [2 2] x1 + [7]
                             [2 0]      [7]
                c_3(x1) = [0 2] x1 + [3]
                          [0 2]      [3]
           
           * Path {4}->{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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6() = [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: {a__c^#() -> c_1()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                a__c^#() = [2]
                           [2]
                c_1() = [1]
                        [0]
                mark^#(x1) = [0 0] x1 + [7]
                             [1 2]      [7]
                c_3(x1) = [2 0] x1 + [2]
                          [2 2]      [3]
           
           * Path {4}->{7}: 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6() = [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: {a__c^#() -> c_6()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                a__c^#() = [2]
                           [2]
                mark^#(x1) = [0 0] x1 + [7]
                             [1 2]      [7]
                c_3(x1) = [2 0] x1 + [2]
                          [2 2]      [3]
                c_6() = [1]
                        [0]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6() = [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: {mark^#(b()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [2]
                      [2]
                mark^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_4() = [0]
                        [1]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: a__f^#(b(), X, c()) -> c_0(a__f^#(X, a__c(), X))
              , 2: a__c^#() -> c_1()
              , 3: mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
              , 4: mark^#(c()) -> c_3(a__c^#())
              , 5: mark^#(b()) -> c_4()
              , 6: a__f^#(X1, X2, X3) -> c_5()
              , 7: a__c^#() -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{3}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                `->{6}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}->{6}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 {3}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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:
                 {  mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
                  , a__f^#(X1, X2, X3) -> c_5()
                  , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                  , mark(c()) -> a__c()
                  , mark(b()) -> b()
                  , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                  , a__c() -> b()
                  , a__f(X1, X2, X3) -> f(X1, X2, X3)
                  , a__c() -> c()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6() = [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: {mark^#(c()) -> c_3(a__c^#())}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                a__c^#() = [2]
                mark^#(x1) = [2] x1 + [5]
                c_3(x1) = [2] x1 + [3]
           
           * Path {4}->{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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6() = [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: {a__c^#() -> c_1()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [0]
                a__c^#() = [2]
                c_1() = [1]
                mark^#(x1) = [0] x1 + [6]
                c_3(x1) = [2] x1 + [2]
           
           * Path {4}->{7}: 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6() = [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: {a__c^#() -> c_6()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [0]
                a__c^#() = [2]
                mark^#(x1) = [0] x1 + [6]
                c_3(x1) = [2] x1 + [2]
                c_6() = [1]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6() = [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: {mark^#(b()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [7]
                mark^#(x1) = [1] x1 + [7]
                c_4() = [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 TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex24 Luc06 GM

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex24 Luc06 GM

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  a__f(b(), X, c()) -> a__f(X, a__c(), X)
     , a__c() -> b()
     , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
     , mark(c()) -> a__c()
     , mark(b()) -> b()
     , a__f(X1, X2, X3) -> f(X1, X2, X3)
     , a__c() -> c()}

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: a__f^#(b(), X, c()) -> c_0(a__f^#(X, a__c(), X))
              , 2: a__c^#() -> c_1()
              , 3: mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
              , 4: mark^#(c()) -> c_3(a__c^#())
              , 5: mark^#(b()) -> c_4()
              , 6: a__f^#(X1, X2, X3) -> c_5(X1, X2, X3)
              , 7: a__c^#() -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{3}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                `->{6}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}->{6}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 {3}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
                  , a__f^#(X1, X2, X3) -> c_5(X1, X2, X3)
                  , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                  , mark(c()) -> a__c()
                  , mark(b()) -> b()
                  , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                  , a__c() -> b()
                  , a__f(X1, X2, X3) -> f(X1, X2, X3)
                  , a__c() -> c()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(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_6() = [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: {mark^#(c()) -> c_3(a__c^#())}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                      [2]
                a__c^#() = [2]
                           [2]
                           [0]
                mark^#(x1) = [0 2 2] x1 + [7]
                             [2 2 0]      [7]
                             [0 0 0]      [7]
                c_3(x1) = [2 2 0] x1 + [1]
                          [0 0 0]      [7]
                          [2 0 0]      [3]
           
           * Path {4}->{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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(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_6() = [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: {a__c^#() -> c_1()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [6]
                      [2]
                a__c^#() = [2]
                           [2]
                           [2]
                c_1() = [1]
                        [0]
                        [0]
                mark^#(x1) = [2 0 2] x1 + [7]
                             [0 2 0]      [3]
                             [2 0 0]      [7]
                c_3(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]
           
           * Path {4}->{7}: 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(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_6() = [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: {a__c^#() -> c_6()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [6]
                      [2]
                a__c^#() = [2]
                           [2]
                           [2]
                mark^#(x1) = [2 0 2] x1 + [7]
                             [0 2 0]      [3]
                             [2 0 0]      [7]
                c_3(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]
                c_6() = [1]
                        [0]
                        [0]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__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]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                a__c() = [0]
                         [0]
                         [0]
                mark(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                a__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]
                a__c^#() = [0]
                           [0]
                           [0]
                c_1() = [0]
                        [0]
                        [0]
                mark^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(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_6() = [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: {mark^#(b()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [2]
                      [2]
                      [2]
                mark^#(x1) = [0 2 0] x1 + [7]
                             [2 2 0]      [3]
                             [2 2 2]      [3]
                c_4() = [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: a__f^#(b(), X, c()) -> c_0(a__f^#(X, a__c(), X))
              , 2: a__c^#() -> c_1()
              , 3: mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
              , 4: mark^#(c()) -> c_3(a__c^#())
              , 5: mark^#(b()) -> c_4()
              , 6: a__f^#(X1, X2, X3) -> c_5(X1, X2, X3)
              , 7: a__c^#() -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{3}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                `->{6}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}->{6}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 {3}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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:
                 {  mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
                  , a__f^#(X1, X2, X3) -> c_5(X1, X2, X3)
                  , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                  , mark(c()) -> a__c()
                  , mark(b()) -> b()
                  , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                  , a__c() -> b()
                  , a__f(X1, X2, X3) -> f(X1, X2, X3)
                  , a__c() -> c()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_6() = [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: {mark^#(c()) -> c_3(a__c^#())}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                a__c^#() = [0]
                           [2]
                mark^#(x1) = [2 2] x1 + [7]
                             [2 0]      [7]
                c_3(x1) = [0 2] x1 + [3]
                          [0 2]      [3]
           
           * Path {4}->{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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_6() = [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: {a__c^#() -> c_1()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                a__c^#() = [2]
                           [2]
                c_1() = [1]
                        [0]
                mark^#(x1) = [0 0] x1 + [7]
                             [1 2]      [7]
                c_3(x1) = [2 0] x1 + [2]
                          [2 2]      [3]
           
           * Path {4}->{7}: 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_6() = [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: {a__c^#() -> c_6()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                a__c^#() = [2]
                           [2]
                mark^#(x1) = [0 0] x1 + [7]
                             [1 2]      [7]
                c_3(x1) = [2 0] x1 + [2]
                          [2 2]      [3]
                c_6() = [1]
                        [0]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                a__c() = [0]
                         [0]
                mark(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                a__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]
                a__c^#() = [0]
                           [0]
                c_1() = [0]
                        [0]
                mark^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_6() = [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: {mark^#(b()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [2]
                      [2]
                mark^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_4() = [0]
                        [1]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: a__f^#(b(), X, c()) -> c_0(a__f^#(X, a__c(), X))
              , 2: a__c^#() -> c_1()
              , 3: mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
              , 4: mark^#(c()) -> c_3(a__c^#())
              , 5: mark^#(b()) -> c_4()
              , 6: a__f^#(X1, X2, X3) -> c_5(X1, X2, X3)
              , 7: a__c^#() -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{3}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                `->{6}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {3}->{1}->{6}.
           
           * Path {3}->{1}->{6}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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 {3}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                , mark(c()) -> a__c()
                , mark(b()) -> b()
                , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                , a__c() -> b()
                , a__f(X1, X2, X3) -> f(X1, X2, X3)
                , a__c() -> c()}
             
             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:
                 {  mark^#(f(X1, X2, X3)) -> c_2(a__f^#(X1, mark(X2), X3))
                  , a__f^#(X1, X2, X3) -> c_5(X1, X2, X3)
                  , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
                  , mark(c()) -> a__c()
                  , mark(b()) -> b()
                  , a__f(b(), X, c()) -> a__f(X, a__c(), X)
                  , a__c() -> b()
                  , a__f(X1, X2, X3) -> f(X1, X2, X3)
                  , a__c() -> c()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6() = [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: {mark^#(c()) -> c_3(a__c^#())}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                a__c^#() = [2]
                mark^#(x1) = [2] x1 + [5]
                c_3(x1) = [2] x1 + [3]
           
           * Path {4}->{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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6() = [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: {a__c^#() -> c_1()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [0]
                a__c^#() = [2]
                c_1() = [1]
                mark^#(x1) = [0] x1 + [6]
                c_3(x1) = [2] x1 + [2]
           
           * Path {4}->{7}: 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(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6() = [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: {a__c^#() -> c_6()}
               Weak Rules: {mark^#(c()) -> c_3(a__c^#())}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [0]
                a__c^#() = [2]
                mark^#(x1) = [0] x1 + [6]
                c_3(x1) = [2] x1 + [2]
                c_6() = [1]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(a__f) = {}, Uargs(mark) = {}, Uargs(f) = {},
                 Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a__f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                b() = [0]
                c() = [0]
                a__c() = [0]
                mark(x1) = [0] x1 + [0]
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                a__f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                a__c^#() = [0]
                c_1() = [0]
                mark^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6() = [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: {mark^#(b()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mark^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [7]
                mark^#(x1) = [1] x1 + [7]
                c_4() = [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.